On the stability of a laminated beam with structural damping and Gurti–Pipkin thermal law

Liu Wenjun; Zhao Weifan

resúmenes

secciones

referencias

imágenes

Abstract: In this paper, we investigate the stabilization of a one-dimensional thermoelastic laminated beam with structural damping coupled with a heat equation modeling an expectedly dissipative effect through heat conduction governed by Gurtin–Pipkin thermal law. Under some assumptions on the relaxation function ., we establish the well-posedness of the problem by using Lumer–Phillips theorem. Furthermore, we prove the exponential stability and lack of exponential stability depending on a stability number by using the perturbed energy method and Gearhart– Herbst–Prüss–Huang theorem, respectively.

Keywords: laminated beam, Gurtin–Pipkin thermal law, well-posedness, exponential stability, lack of exponential stability.

Carátula del artículo

Article

On the stability of a laminated beam with structural damping and Gurti–Pipkin thermal law

Liu Wenjun wjliu@nuist.edu.cn

Nanjing University of Information Science and Technology,, China

Zhao Weifan

Nanjing University of Information Science and Technology, China

Nonlinear Analysis: Modelling and Control, vol. 26, núm. 3, pp. 396-418, 2021
Vilniaus Universitetas

Esta obra está bajo una Licencia Creative Commons Atribución 4.0 Internacional.

Recepción: 24 Noviembre 2019

Revisado: 16 Noviembre 2020

Publicación: 01 Mayo 2021

1 Introduction

In this paper, we investigate the well-posedness and asymptotic stability of a thermoelastic laminated beam with structural damping and Gurtin–Pipkin thermal law, i.e., for (x, t) ∈ (0, 1) × (0, +∞),

(1)

with the initial and boundary conditions

(21)

(22)

where the functionsdenote the trans- verse displacement of the beam, which departs from its equilibrium position, rotation angle, effective rotation angle, relative temperature, and the memory kernel, respectively; .w (x, t) is proportional to the amount of slip along the interface at time t and longitudinal spatial variable x; g(s) is the heat conductivity relaxation kernel, whose properties will be specified later; (1). describes the dynamics of the slip; ρ, G, I., D, γ, βare the density of the beams, shear stiffness, mass moment of inertia, flexural rigidity, adhesive stiffness of the beams, and adhesive damping parameter, respectively. Moreover,ρ, G, I., D, δ, γ, α, k, βare positive constant.

Problem (1) is closely related to 1D thermoelastic Timoshenko beam model in the sense that (1) reduces to the Timoshenko system with Gurtin–Pipkin thermal law [12] studied by Dell’Oro and Pata [9] if the slip . is assumed to be identically zero. When there is no thermal effect, problem (1) is called laminated beam. Hansen [13] derived a model for a two-layered plate in which slip could occur along the interface. Concerned with the beam analog, with strain-rate damping as in the above described plate model [13, Eq. (3.16)], the basic evolution equations for the system are given by

where . is the shear force, and . is the bending moment. The constitutive equations are Hansen and Spies [14] derived the mathematical model for two-layered beams with structural damping due to the interfacial slip, namely,

(3)

for (x, t) ∈ (0, 1) × (0, +∞). Later on, Wang et al. [29] consi√dered syste√m (3) with the cantilever boundary conditions and two different wave speeds they pointed out that system (3) can reach the asymptotic stability, but it does not reach the exponential stability due to the action of the slip w. To achieve the exponential decay result, the authors in [29] added an additional boundary control such that the boundary conditions become

where are positive constant feedback gains. Furthermore, Cao et al. [3] proved the exponential stability for system (3) with following boundary conditions:

provided . More importantly, the authors proved that the dominant part of the system is itself exponentially stable.

Concerning a laminated beam with thermoelastic dissipation effective in the bending moment, we have

(4)

where 0 is the temperature difference, q denotes the heat flux, S = G(ψ - ϕ_x), and M = D(3w - ψ)_x δθ. Derivative of the heat flux term in the formulation of the rate equation

(5)

was introduced independently by Cattaneo [4] and Vernotte [28] with a fixed constant κ > 0 and small τ > 0. Combining (4) and (5), Apalara [1] considered a laminated beam with structural damping and second sound

(6)

for The stabilization of system (6) has been analyzed in [1], where Apalara obtained the well-posedness and uniform stability results depending on the following stability number:

Mukiawa et al. [23] studied a thermoelastic laminated beam system without structural damping, but with a finite memory acting on the bending moment and established a gen- eral and optimal decay estimate. If we assume Gurtin–Pipkin thermal law [12] of heat conduction

(7)

where g is called the memory kernel, we can obtain equation (1).. The aim of this paper is to study the well-posedness and asymptotic stability of a thermoelastic laminated beam with structural damping and Gurtin–Pipkin thermal law, i.e., (1)–(2). In fact, Cattaneo law (5) can be reduced as a particular instance of (7), which have been proved in [9]. For other asymptotic behavior results to laminated beams, we refer the reader to [6, 14, 16, 21, 29] and the references therein.

For the case of the beams with Gurtin–Pipkin thermal law [12], a large number of interesting decay results depending on the stability number have been established. Re- cently, Dell’Oro and Pata [9] considered Timoshenko system with Gurtin–Pipkin thermal law, i.e., for (x, t) ∈ (0, L) × (0, +∞),

where are positive constants. The authors obtained the exponential stability depending on the stability number

Later on, Dell’Oro [8] considered the thermoelastic Bresse–Gurtin–Pipkin system, i.e., for (x, t) ∈ (0, L) × (0, +∞),

and obtained that the system is exponentially stable if and only if

For other related results, we refer the reader to [5, 17–20, 26].

In this paper, we first prove the well-posedness by using Lumer–Phillips theorem. And then, by using the perturbed energy method, we establish an exponential stability result depending on the stability number

To overcome the difficulty brought by Gurtin–Pipkin thermal law, we use some appropri- ated multipliers to construct a Lyapunov functional. For the case we prove the lack of exponential stability by using Gearhart–Herbst–Prüss–Huang theorem.

The remaining part of this paper is organized as follows. In Section 2, we introduce some hypotheses and present our main results. In Section 3, we prove the well-posedness for problem (1)–(2). In Section 4, we establish an exponential decay result to prob- lem (1)–(2). In Section 5, we prove the lack of exponential stability for problem (1)–(2). Section 6 is devoted to the conclusion and open problem. Throughout this paper, we use . to denote a generic positive constant.

2 Preliminaries and main results

In this section, we first introduce some notation and present our hypotheses. Then we give some lemmas, which will be used in the proof of main results.

To deal with the memory, following [7], we introduce a new auxiliary variable by (see also [9, 10])

which satisfies the boundary conditions Then . satisfies where . and Assume g(∞) = 0, a change of variable and a formal integration by parts yield

Now, we denote then Hence system (1)–(2) can be written

(8)

for (x, t) ∈ (0, 1) × (0, +∞) with initial and boundary conditions

(91)

(92)

For the memory kernel ., we assume and

(G1) g is a bounded convex summable function on [0, ∞);

(G2) g has a total mass

(G3) g´. is an absolutely continuous function on R+ so that

(G4) There exists a positive constant so that, for almost every s > 0,

Remark 1.

In particular, u is summable on R+ with Furthermore, noting that g(s) has total mass 1 , we have

Next, we introduce the vector function with Then system (8)–(9) can be written as

(10)

where A is a linear operator defined by

We consider the following spaces:

and the energy space

where

equipped with the normand inner product In particular

Moreover, in light of (G4) on u, we deduce

(11)

Besides, H is the Hilbert space equipped with the norm

and the inner product

for . =(ϕ, u, 3.−ψ, 3.−u, w, v, θ, η). and .˜ =(.˜, u˜, 3.˜−.˜, 3.˜−.˜, w., v., θ., η˜).

The domain of A is given by

where Clearly, D( A) is dense in H.

The energy associated with problem (8)–(9) is defined by

(12)

Now, we give our main results in this paper as follows.

Theorem 1.

Let U₀ ∈H, then problem (10)admits a unique weak solution U ∈ C(R+; H). Moreover, if U₀∈ D(A), then .

Theorem 2.

Assume that then there exist positive constants a, b such that the energy E(t)associated with problem (8).(9) satisfies

(13)

Theorem 3.

Assume that then problem (8).(9) is not exponentially stable.

Based on two propositions from [9, Props. 11, 12], we give the full equivalence between Cattaneo law and Gurtin–Pipkin thermal law.

Theorem 4.

If the laminated beam with structural damping and Cattaneo law is exponen- tially stable, then so is the laminated beam with structural damping and Gurtin–Pipkin thermal law, and vice versa.

3 Well-posedness: proof of Theorem 1

To obtain the well-posedness, we need to prove that is a maximal mono- tone operator. To achieve this goal, we need to prove that A is dissipative and Id - Ais surjective.

Using the inner product and integration by parts, we can easily obtain

for any U ∈ D(A). Hence A is dissipative.

Next, we turn to prove Id − A is surjective, i.e., for any there exists satisfying

(14)

that is,

(15)

From (15)₁and v_S(0) = 0 we have

(16)

Inserting (16) into (15)2, (15)3, (15)4 and (15)5, we obtain

(17)

Multiplying (17) by, and, respectively, and integrating over (0, 1), we can obtain

(18)

From (18) we have the following variational formulation:

(19)

for all where

and

Now, we introduce the Hilbert space equipped with the norm

Then and are bounded. Furthermore, we obtain that there exists a positive constant . such that

Hence is coercive.

As a consequence, by applying Lax–Milgram lemma [24] we can obtain that (18) has a unique solutionThen, substituting into (16)1, we obtain

Using (16)2 and the method in [30, Prop. 2.2], we have

which gives us Then from (15). we can obtain.

Hence,. Next, we turn to prove that

Now, ifthen (19) reduces to

(20)

for allwhich implies

(21)

From the regularity theory for the linear elliptic equations, we obtain

Moreover, (20) is also true for any Thus, we get

for all, φ(1) = 0. Using (21) and the integration by parts, we have

Hence,In the same way, we get

From the classical regularity theory for the linear elliptic equations we know that there exists a unique solution such that (14) is satisfied. So the operator Id is surjective.

As a consequence, A is a maximal monotone operator. Therefore, we established the well-posedness result stated in Theorem 1 by using Lumer–Phillips theorem (see [2]).

4 Exponential decay: proof of Theorem 2

In this section, we prove the exponential stability for system (8)–(9) when χ_g = 0. It will be achieved by using the perturbed energy method. Before we prove our result, we need some useful lemmas.

Lemma 1.

Let (ϕ, ψ, w, θ) be the solution of problem (8).(9). Then the energy function

E(t)satisfies

(22)

Proof. Multiplying (8)1 by, (8)2 by (3w ψ)_t, (8)3 by 3w_t, (8)4 by 0 and integrat- ing over (0, 1), using integration by parts and the boundary conditions in (9), we can obtain (22). This completes the proof.

Lemma 2.

Let (ϕ, ψ, w, θ) be the solution of (8).(9). Then the functional satisfies the estimate

(23)

for any ε₁> 0.

Proof. Taking the derivative of F1(t) with respect to t, using (8)4, (8)5 and integrating by parts, we get

Using integration by parts, Poincaré’s inequality [27, Lemma 2.2], and Young’s inequality with, we infer that

Here we take then we can get (23) by using above inequalities and (11). This completes the proof.

Lemma 3.

Let (ϕ, ψ, w, θ) be the solution of (8).(9). Then the functional satisfies the estimate

(24)

for any ε₂ > 0.

Proof. Taking the derivative of F2(t) with respect to t, using (8)2, (8)4 and integrating by parts, we get

Using (11), Young’s and Cauchy–Schwarz inequalities with we establish esti- mate (24).

Lemma 4.

Let (ϕ, ψ, w, θ) be the solution of (8).(9). Then the functional

satisfies the estimat

(25)

Proof. By (8)1, (8)2, (8)4 and integrating by parts, we get

Similarly as in [1, Lemma 2.4], using χ_g= 0, Young’s and Cauchy–Schwarz inequalities, and the fact that we get (25).

Lemma 5.

Let (ϕ, ψ, w, θ) be the solution of (8).(9). Then the functional satisfies the estimate

(26)

for any ε₄> 0.

Proof. By differentiating F4(t) with respect to t, using (8). and integrating by parts, we obtain

Using Young’s and Poincaré’s inequalities, we obtain

for Note that

Then estimate (26) is obtained.

Lemma 6.

Let (ϕ, ψ, w, θ) be the solution of (8).(9). Then the functional satisfies the estimate

(27)

Proof. Taking the derivative of F₅(t) with respect to t, using (8)2 and integrating by parts, we get

Then, using Poincaré’s and Young’s inequalities, we arrive at (27).

Lemma 7.

Let (ϕ, ψ, w, θ) be the solution of (8).(9). Then the functional satisfies the estimate

(28)

Proof. By differentiating F₆(t) with respect to t, using (8)3 and integrating by parts, then use Young’s inequality to obtain (28). This completes the proof.

Now we define the following Lyapunov functional

where are positive constants to be selected later. Then we have the lemma as follows.

Lemma 8.

Let (ϕ, ψ, w, θ) be the solution of (8).(9). For N large enough, there exists a positive c such that, for any t ≥ 0,

Proof. Using Young’s, Poincaré’s and Cauchy–Schwarz inequalities, and the fact that (see [22])

we can easily obtain that

(29)

whereα_i(i = 1,2, . . . , 9) are positive constants. It follows from (12) and (29) that there exists a positive constant . such that which completes the proof.

Now, we are ready to prove the main result in this section.

Proof of Theorem 2. From (23)–(27) and (28) we can obtain

At this point, we need to choose our constants very carefully. First, we choose so that

(30)

Then, we select .. large enough so that Next, we select .. large enough so that. Furthermore, we select .. large enough so that Finally, we select . large enough so that Using (12), we obtain that there exist positive constants M1 and M2 such that (30) becomes

From Lemma 8 we obtain

(31)

where Then, a simple integration of (31) over (0, t) yields

(32)

At last, estimate (32) gives exponential stability result (13) when be combined with

Lemma 8. This completes the proof.

5 Lack of exponential stability: proof of Theorem 3

Our result is achieved by using Gearhart–Herbst–Prüss–Huang theorem to dissipative systems (see Prüss [25] and Huang [15]).

Lemma 9.

-semigroup of contractions on Hilbert space H Then S(t) is exponentially stable if and only if

hold, where ρ(A) is the resolvent set of the differential operator A.

Proof of Theorem 3. We will prove that there exists a sequence of imaginary number λ. and function Fu ∈ H with ǁF_uǁ_H ≤ 1 such that where

(33)

with not bounded. Rewriting spectral equation (33) in term of its components, we have for,

where Take then the above system becomes

Due to the boundary conditions in (9), we can suppose that .. Choosing

then we can obtain

(34)

In the above equations, we take Solving (34)5, we get

(35)

Then substituting (35) into (34)4, we can get

The combination of (34)2 and (34)3 gives

(36)

Substituting . into (36), we get . where

Substituting C into (34)1, we get

Similarly, substituting C into (34)3, we get

At this point, we introduce the number and consider separately two cases.

Case we get

Thus,

This implies that Therefore, there is no exponential stability. This completes the proof.

6 Conclusion and open problem

In this paper, we first prove the well-posedness for a laminated beam with Gurtin–Pipkin thermal law and structural damping. Then we prove that the system is exponentially stable if and only if that stability number is equal to zero (χ_g = 0). When the stability number is not zero, the problem of whether it is possible to get the polynomial stability for system (8)–(9) is still an interesting open problem.

Recently, Guesmia [11] considered the stability of the laminated beam with interfacial slip and infinite memory acting only on the transverse displacement, the rotation angle, and the amount of slip, respectively. He combined the energy method and the frequency domain approach to show that the infinite memory is capable alone to guarantee the strong and polynomial stability of the model, and mentioned also that “when the exponential stability is not satisfied, obtaining the optimal decay rate of solutions is, in our opinion, a very nice and hard question”.

Material suplementario

References

1. T.A. Apalara, Uniform stability of a laminated beam with structural damping and second sound, Z. Angew. Math. Phys., 68(2):41, 2017, https://doi.org/10.1007/s00033-017- 0784-x.

2. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011, https://doi.org/10.1007/978-0-387- 70914-7.

3. X.-G. Cao, D.-Y. Liu, G.-Q. Xu, Easy test for stability of laminated beams with structural damping and boundary feedback controls, J. Dyn. Control Syst., 13(3):313–336, 2007, https://doi.org/10.1007/s10883-007-9022-8.

4. C. Cattaneo, Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, C. R. Acad. Sci., Paris, 247:431–433, 1958.

5. M.M. Chen, W.J. Liu, W.C. Zhou, Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms, Adv. Nonlinear Anal., .(4):547–569, 2018, https://doi.org/10.1515/anona-2016- 0085.

6. Z.J. Chen, W.J. Liu, D.Q. Chen, General decay rates for a laminated beam with memory, Taiwanese J. Math., 23(5):1227–1252, 2019, https://doi.org/10.11650/tjm/ 181109.

7. C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37:297– 308, 1970, https://doi.org/10.1007/BF00251609.

8. F. Dell’Oro, Asymptotic stability of thermoelastic systems of Bresse type, J. Differ. Equations, 258(11):3902–3927, 2015, https://doi.org/10.1016/j.jde.2015.01.025.

9. F. Dell’Oro, V. Pata, On the stability of Timoshenko systems with Gurtin–Pipkin thermal law, J. Differ. Equations, 257(2):523–548, 2014, https://doi.org/10.1016/j.jde. 2014.04.009.

10. C. Giorgi, V. Pata, A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA, Nonlinear Differ. Equ. Appl., .(3):333–354, 1998, https://doi.org/10.1007/s000300050049.

11. A. Guesmia, Well-posedness and stability results for laminated Timoshenko beams with interfacial slip and infinite memory, IMA J. Math. Control Inform., 37(1):300–350, 2020, https://doi.org/10.1093/imamci/dnz002.

12. M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 31(2):113–126, 1968, https://doi.org/10.1007/ BF00281373.

13. S.W. Hansen, A model for a two-layered plate with interfacial slip, in W. Desch, F. Kappel, K. Kunisch (Eds.), Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena. International Conference in Vorau (Austria), July 18–24,1993, ISNM, Int. Ser. Numer. Math., Vol. 118, Birkhäuser, Basel, 1994, pp. 143–170.

14. S.W. Hansen, R. Spies, Structural damping in a laminated beams due to interfacial slip, J. Sound Vib., 204(2):183–202, 1997, https://doi.org/10.1006/jsvi.1996. 0913.

15. F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differ. Equations, .(1):43–56, 1985.

16. G. Li, X.Y. Kong, W.J. Liu, General decay for a laminated beam with structural damping and memory: The case of non-equal wave speeds, J. Integral Equations Appl., 30(1):95–116, 2018, https://doi.org/10.1216/JIE-2018-30-1-95.

17. W.J. Liu, K.W. Chen, J. Yu, Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term, IMA J. Math. Control Inform., 34(2):521–542, 2017, https://doi.org/10.1093/imamci/dnv056.

18. W.J. Liu, K.W. Chen, J. Yu, Asymptotic stability for a non-autonomous full von Kármán beam with thermo-viscoelastic damping, Appl. Anal., 97(3):400–414, 2018, https://doi.org/ 10.1080/00036811.2016.1268688.

19. W.J. Liu, X.Y. Kong, G. Li, Asymptotic stability for a laminated beam with structural damping and infinite memory, Math. Mech. Solids, 25(10):1979–2004, 2020, https://doi.org/ 10.1177/1081286520917440.

20. W.J. Liu, X.Y. Kong, G. Li, Lack of exponential decay for a laminated beam with structural damping and second sound, Ann. Pol. Math., 124(3):281–289, 2020, https://doi.org/ 10.4064/ap181224-17-9.

21. W.J. Liu, W.F. Zhao, Stabilization of a thermoelastic laminated beam with past history, Appl. Math. Optim.,80(1):103–133, 2019, https://doi.org/10.1007/s00245- 017-9460-y.

22. A. Lo, N. Tatar, Uniform stability of a laminated beam with structural memory, Qual. Theory Dyn. Syst., 15(2):517–540, 2016, https://doi.org/10.1007/s12346-015-0147- y.

23. S.E. Mukiawa, T.A. Apalara, S. Messaoudi, A general and optimal stability result for a lam- inated beam, J. Integral Equations Appl., 32(3):341–359, 2020, https://doi.org/10. 1216/jie.2020.32.341.

24. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., Vol. 44, Springer, New York, 1983, https://doi.org/10.1007/ 978-1-4612-5561-1.

25. J. Prüss, On the spectrum of .0-semigroups, Trans. Am. Math. Soc., 284(2):847–857, 1984, https://doi.org/10.2307/1999112.

26. H.D. Fernández Sare, R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194(1):221–251, 2009, https://doi.org/ 10.1007/s00205-009-0220-2.

27. Z. Tian, G.-Q. Xu, Exponential stability analysis of Timoshenko beam system with boundary delays, Appl. Anal., 96(15):2575–2603, 2017, https://doi.org/10.1080/ 00036811.2016.1232399.

28. P. Vernotte, Les paradoxes de la théorie continue de l’équation de la chaleur, C. R. Acad. Sci., Paris, 246:3154–3155, 1958.

29. J.-M. Wang, G.-Q. Xu, S.-P. Yung, Exponential stabilization of laminated beams with structural damping and boundary feedback controls, SIAM J. Control Optim., 44(5):1575–1597, 2005, https://doi.org/10.1137/040610003.

30. Q. Zhang, Stability analysis of an interactive system of wave equation and heat equation with memory, Z. Angew. Math. Phys., 65(5):905–923, 2014, https://doi.org/10.1007/ s00033-013-0366-5.

Notas